Reciprocity law: Difference between revisions

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In mathematics, the '''restricted product''' is a construction in the theory of [[topological group]]s.
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Let <math> I </math> be an indexing set; <math> S </math> a finite subset of <math>I</math>. If for each <math>i\in I</math>, <math> G_i </math> is a [[locally compact]] group, and for each <math>i\in I\backslash S</math>, <math> K_i \subset G_i</math> is an open [[Compact group|compact]] subgroup, then the restricted product
: <math> {\prod_i}' G_i\, </math>
is the subset of the product of the <math> G_i </math>'s consisting of all elements <math> (g_i)_{i\in I}</math> such that <math> g_i \in K_i </math> for all but finitely many <math> i\in I\backslash S</math>.  
 
This group is given the topology whose basis of open sets are those of the form
: <math> \prod_i A_i\,, </math>
where <math>A_i</math> is open in <math>G_i</math> and <math>A_i = K_i</math> for all but finitely many <math>i</math>.
 
One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the [[adele ring]] and [[idele group]] of a [[global field]].
==See also==
*[[Direct sum]]
==References==
 
* {{Citation | last1=Fröhlich | first1=A. | last2=Cassels | first2=J. W. | author2-link=John William Scott Cassels | title=Algebraic number theory | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-163251-9 | year=1967}}
* {{Neukirch ANT}}
 
{{DEFAULTSORT:Restricted Product}}
[[Category:Topological groups]]

Revision as of 10:19, 1 March 2014

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